How to call F6 menu after calling solidify mesh command?
Since you are wrapping the Solidify operator inside your operator :


create a property (variable ) to hold the thickness and pass it to the solidify operator.
create a draw() function and call it ( the F6 will also call this draw() function )
you can display the menu of your operator to control the solidify operator through the property.




import bpy
from bpy.props import FloatProperty


class SolidifyDialog(bpy.types.Operator):  
    bl_idname = "solidify.dialog"  
    bl_label = "Solidify Dialog"  
    bl_options = {'REGISTER', 'UNDO'}

    thickness = FloatProperty(
                name = "thickness",
                default = 0.001,
                min = -10,
                max = 10,
                unit ='LENGTH'   
                )
    def execute(self, context):
        bpy.ops.mesh.solidify(thickness = self.thickness)
        return {'FINISHED'} 

    def draw(self, context):
         layout = self.layout
         row = layout.row()
         row.prop(self,"thickness")

    def invoke(self, context, event):
        wm = context.window_manager
        return wm.invoke_props_popup(self, event)

def register():
    bpy.utils.register_class(SolidifyDialog)


def unregister():
    bpy.utils.unregister_class(SolidifyDialog)


if __name__ == "__main__":
    register()


note : wm.invoke_props_dialog() didn't work as expected , it doesn't update the view while changing the value so i switched to wm.invoke_props_popup()